Melt control: charge optimization via stochastic programming.

*(English)*Zbl 1190.90063
Wallace, Stein W. (ed.) et al., Applications of stochastic programming. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA: MPS, Mathematical Programming Society (ISBN 0-89871-555-5/pbk). MPS/SIAM Series on Optimization 5, 277-297 (2005).

The melt control is considered as a linear programming problem. The goal is to find amounts of the input materials in the lowest cost so that the prescribed output alloy composition is achieved.

The production consists of several steps (e.g. charge, alloying). During the alloying process, the hot melt in the furnace is enriched with input materials, and the new mixture is melted again. Hence, the problem has a natural multistage decision structure. In the article two simplified models are considered: two-stage induced furnace charge optimization and three stage electric-arc furnace charge optimization.

For example, two-stage induced furnace charge optimization problem is \[ \min \left( \sum_{j \in J_1}c_j x_j^1 + \sum_{k_2 \in K_2}p_{k_2}\sum_{j \in J_2} c_j x_j^{k_2} \right) \] subject to \[ l_{i1} \leq \sum_{l=1}^{m_1} \tau_{il}^E \sum_{j\in J_1} a_{lj} x_j^1 \leq u_{i1}, \;i=1,\dots,m_1, \] \[ l_{i2} \leq \sum_{l=1}^{m_1} \tau_{il}^{k_2} \sum_{j\in J_1} a_{lj} x_j^1 + \sum_{j\in J_2} a_{ij} x_j^{k_2} \leq u_{i2}, \;i=1,\dots,m_2, \;k_2 \in K_2, \] \[ x_j^1 \geq 0, \;j\in J_1, \;x_j^{k_2} \geq 0, \;j\in J_2, \;k_2 \in K_2, \] where \(J_t\) is the set of indices available at stage \(t\) (\(t=1,2\)); \(m_t\), is the number of elements at stage \(t\); \(c_j \geq 0\), \(j \in J_t\) are known costs of input materials; \(l_{it}\), \(u_{it}\) are lower and upper bounds of the \(i\)th element in the melt composition at stage \(t\); \(a_{ij}\geq 0\) denotes the amount of the \(i\)th element in the unit amount of the \(j\)th input material; \(x_j^1\), \(x_j^{k_2}\) are the amounts of the \(j\)th input material at the first and second stages under scenario \(k_2\); \(\tau_{il}^{k_2}\) is utilization of the \(i\)th element related to the amount of the \(l\)th element in the melt when scenario \(k_2\) occurs; \(k_2\) are indices of scenarios with probabilities \(p_{k_2} \geq 0\).

The scenario generation, which is one of the most important tasks, is proposed as well. Then two-stage melt control and three-stage melt control problems based on real-life data are considered. Prospective extensions of the method are discussed.

For the entire collection see [Zbl 1068.90002].

The production consists of several steps (e.g. charge, alloying). During the alloying process, the hot melt in the furnace is enriched with input materials, and the new mixture is melted again. Hence, the problem has a natural multistage decision structure. In the article two simplified models are considered: two-stage induced furnace charge optimization and three stage electric-arc furnace charge optimization.

For example, two-stage induced furnace charge optimization problem is \[ \min \left( \sum_{j \in J_1}c_j x_j^1 + \sum_{k_2 \in K_2}p_{k_2}\sum_{j \in J_2} c_j x_j^{k_2} \right) \] subject to \[ l_{i1} \leq \sum_{l=1}^{m_1} \tau_{il}^E \sum_{j\in J_1} a_{lj} x_j^1 \leq u_{i1}, \;i=1,\dots,m_1, \] \[ l_{i2} \leq \sum_{l=1}^{m_1} \tau_{il}^{k_2} \sum_{j\in J_1} a_{lj} x_j^1 + \sum_{j\in J_2} a_{ij} x_j^{k_2} \leq u_{i2}, \;i=1,\dots,m_2, \;k_2 \in K_2, \] \[ x_j^1 \geq 0, \;j\in J_1, \;x_j^{k_2} \geq 0, \;j\in J_2, \;k_2 \in K_2, \] where \(J_t\) is the set of indices available at stage \(t\) (\(t=1,2\)); \(m_t\), is the number of elements at stage \(t\); \(c_j \geq 0\), \(j \in J_t\) are known costs of input materials; \(l_{it}\), \(u_{it}\) are lower and upper bounds of the \(i\)th element in the melt composition at stage \(t\); \(a_{ij}\geq 0\) denotes the amount of the \(i\)th element in the unit amount of the \(j\)th input material; \(x_j^1\), \(x_j^{k_2}\) are the amounts of the \(j\)th input material at the first and second stages under scenario \(k_2\); \(\tau_{il}^{k_2}\) is utilization of the \(i\)th element related to the amount of the \(l\)th element in the melt when scenario \(k_2\) occurs; \(k_2\) are indices of scenarios with probabilities \(p_{k_2} \geq 0\).

The scenario generation, which is one of the most important tasks, is proposed as well. Then two-stage melt control and three-stage melt control problems based on real-life data are considered. Prospective extensions of the method are discussed.

For the entire collection see [Zbl 1068.90002].

Reviewer: Alex V. Kolnogorov (Novgorod)